DOI: 10.2478/s12175-013-0152-z Math. Slovaca 63 (2013), No. 5, 1025–1036 APPROXIMATION BY COMPLEX SUMMATION-INTEGRAL TYPE OPERATOR IN COMPACT DISKS Vijay Gupta — Rani Yadav (Communicated by J´ an Bors´ ık ) ABSTRACT. In the present paper we estimate a Voronovskaja type quantitative estimate for a certain type of complex Durrmeyer polynomials, which is different from those studied previously in the literature. Such estimation is in terms of analytic functions in the compact disks. In this way, we present the evidence of overconvergence phenomenon for this type of Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane. In the end, we mention certain applications. c 2013 Mathematical Institute Slovak Academy of Sciences 1. Inroduction If f : G → C is an analytic function in the open set G ⊂ C, with D 1 ⊂ G (where D 1 = {z ∈ C : |z | < 1}), then S. N. Bernstein proved that the complex Bernstein polynomials converges uniformly to f in D 1 (see e.g., Lorentz [8: p. 88]). Sorin G Gal has done commendable work in this direction and he estimated upper quantitative estimates for the uniform convergence for the first time. (see e.g. [3: p. 264]). Also exact quantitative estimates for different operators were established in his recent papers see e.g. [2], [4], [6] and [5] etc. In the recent years and for the real variable case, Abel-Gupta-Mohapatra [1] studied the rate of convergence and established asymptotic expansion of certain Bernstein-Durrmeyer type operators, which are discretely defined at f (0). The aim of the present article is to extend the studies on such operators. Let R> 1 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 30E10; Secondary 41A25. K e y w o r d s: complex Durrmeyer-type operators, uniform convergence, analytic function, Voronovskaja-type result.
Transcript
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DOI: 10.2478/s12175-013-0152-z
Math. Slovaca 63 (2013), No. 5, 1025–1036
APPROXIMATION BY COMPLEX
SUMMATION-INTEGRAL TYPE OPERATOR
IN COMPACT DISKS
Vijay Gupta — Rani Yadav
(Communicated by Jan Borsık )
ABSTRACT. In the present paper we estimate a Voronovskaja type quantitativeestimate for a certain type of complex Durrmeyer polynomials, which is differentfrom those studied previously in the literature. Such estimation is in terms ofanalytic functions in the compact disks. In this way, we present the evidence
of overconvergence phenomenon for this type of Durrmeyer polynomials, namelythe extensions of approximation properties (with quantitative estimates) fromreal intervals to compact disks in the complex plane. In the end, we mentioncertain applications.
If f : G → C is an analytic function in the open set G ⊂ C, with D1 ⊂ G(where D1 = {z ∈ C : |z| < 1}), then S. N. Bernstein proved that the complexBernstein polynomials converges uniformly to f in D1 (see e.g., Lorentz [8:p. 88]). Sorin G Gal has done commendable work in this direction and heestimated upper quantitative estimates for the uniform convergence for the firsttime. (see e.g. [3: p. 264]). Also exact quantitative estimates for differentoperators were established in his recent papers see e.g. [2], [4], [6] and [5] etc.
In the recent years and for the real variable case, Abel-Gupta-Mohapatra [1]studied the rate of convergence and established asymptotic expansion of certainBernstein-Durrmeyer type operators, which are discretely defined at f(0). Theaim of the present article is to extend the studies on such operators. Let R > 1
2010 Mathemat i c s Sub j e c t C l a s s i f i c a t i on: Primary 30E10; Secondary 41A25.Keywords: complex Durrmeyer-type operators, uniform convergence, analytic function,Voronovskaja-type result.
VIJAY GUPTA — RANI YADAV
and suppose that f : DR → C is analytic in DR = {z ∈ C : |z| < R} that is
we can write f(z) =∞∑k=0
ckzk, for all z ∈ DR, the complex Bernstein-Durrmeyer
type operator defined as
Mn(f, z) = (n+ 1)
n∑k=1
pn,k(z)
1∫0
f(t)pn,k−1(t) dt+ f(0)pn,0(z) (1)
where z ∈ C, n = 1, 2, . . . and
pn,k(z) :=
(n
k
)zk(1− z)n−k.
Our results will put in evidence the overconvergence phenomenon for the op-erators (1). The results established here are the extensions of approximationproperties with exact quantitative estimates from the real interval [0, 1], to com-pact disks in the complex plane. Also, the methods used here are different fromother complex Bernstein-type operators.
2. Basic results
In the sequel, we shall need the following basic results.
����� 1� For all ep = tp, p ∈ N ∪ {0} and z ∈ C we have
Mn(ep+1, z) =z(1− z)
n+ p+ 2M ′
n(ep, z) +nz + p
n+ p+ 2Mn(ep, z).
P r o o f. For p = 0 the relationship is evident fromMn(e0, z)=1 andMn(e1, z) =nzn+2 (see e.g., [1]).
Therefore, let p ∈ N. Using the equality
z(1− z)p′n,k(z) = (k − nz)pn,k(z),
we have
z(1− z)M ′n(ep, z)
= (n+ 1)
n∑k=1
z(1− z)p′n,k(z)
1∫0
pn,k−1(t)tp dt
= (n+ 1)
n∑k=1
(k − nz)pn,k(z)
1∫0
pn,k−1(t)tp dt
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APPROXIMATION BY COMPLEX SUMMATION-INTEGRAL TYPE OPERATOR
then it is obvious that Ek,n(z) is a polynomial of degree less than or equal to kand by simple computation and the use of above recurrence relation, we are ledto
Ek,n(z) =z(1− z)
n+ k + 1E′
k−1,n(z) +nz + k − 1
n+ k + 1Ek−1,n(z) +Xk,n(z),
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VIJAY GUPTA — RANI YADAV
where
Xk,n(z) =zk−2
n(n+ k + 1)
[z2(2k3 + 2k) + z(−4k3 + 9k2 − 9k + 4)
+(2k3 − 9k2 + 13k − 6)],
for all k ≥ 1, n ∈ N and |z| ≥ r.
Using the estimate in the proof of Theorem 1, we have
|πk,n(z)− ek(z)| ≤ 2k(k + 1)rk
n,
for all k, n ∈ N, |z| ≤ r, with 1 ≤ r.
For all k, n ∈ N, k ≥ 1 and |z| ≤ r, it follows
|Ek,n(z)| ≤ r(1 + r)
n+ k + 1|E′
k−1,n(z)|+nr + k − 1
n+ k + 1|Ek−1,n(z)|+ |Xk,n(z)|.
Since r(1+r)n+k+1 ≤ r(1+r)
n and nr+k−1n+k+1 ≤ r, it follows
|Ek,n(z)| ≤ r(1 + r)
n|E′
k−1,n(z)|+ r|Ek−1,n(z)|+ |Xk,n(z)|.Now we shall find the estimation of |E′
k−1,n(z)| for k ≥ 1. Taking into account
the fact that Ek−1,n(z) is a polynomial of degree ≤ k − 1, we have
|E′k−1,n(z)| ≤
k − 1
r‖Ek−1,n‖r
≤ k − 1
r
[‖πk−1,n − ek−1‖r +
∥∥∥∥ (k − 1)ek−2[(k − 2)− ke1]
n
∥∥∥∥r
]
≤ k − 1
r
[2(k − 1)krk−1
n+
rk−2(k − 1)k(1 + r)
n
]
≤ (k − 1)2k
n
[2rk−1 +
1 + r
rrk−1
]≤ 4(k − 1)2krk−1
n.
Thusr(1 + r)
n|E′
k−1,n(z)| ≤4(k − 1)2k(1 + r)rk
n2
and
|Ek,n(z)| ≤ 4(k − 1)2k(1 + r)rk
n2+ r|Ek−1,n(z)|+ |Xk,n(z)|,
where
|Xk,n(z)| ≤ rk−2
n2
[r2(2k3 + 2k) + r(4k3 + 9k2 + 9k + 4)
+(2k3 + 9k2 + 13k + 6)] ≤ rk
n2Ak,r,
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APPROXIMATION BY COMPLEX SUMMATION-INTEGRAL TYPE OPERATOR
where Bk,r is a polynomial of degree 3 in k defined as
Bk,r = Ak,r + 4(k − 1)2k(1 + r).
But E0,n(z) = 0, for any z ∈ C and therefore by writing last inequality fork = 1, 2, . . . we easily obtain step by step the following
|Ek,n(z)| ≤ rk
n2
k∑j=1
Bj,r ≤ krk
n2Bk,r.
We conclude that∣∣∣∣Mn(f, z)− f(z)− z(1− z)f ′′(z)− 2zf ′(z)n
∣∣∣∣≤
∞∑k=1
|ck||Ek,n| ≤ 1
n2
∞∑k=1
|ck|kBk,rrk.
As f (4)(z) =∞∑k=4
ckk(k−1)(k−2)(k−3)zk−4 and the series is absolutely conver-
gent in |z| ≤ r, it easily follows that∞∑k=4
|ck|k(k−1)(k−2)(k−3)rk−4 < ∞, which
implies that∞∑k=1
|ck|kBk,rrk < ∞. This completes the proof of theorem. �
Finally, we will obtain the exact order in approximation by this type of com-plex Bernstein-Durrmeyer polynomials and by their derivatives. In this sense,we present the following result.
������ 3� Let R > 1 and suppose that f : Dr → C is analytic in DR, that
is we can write f(z) =∞∑k=0
ckzk, for all z ∈ DR. If f is a polynomial of degree
Taking into account that by hypothesis f is not a polynomial of degree 0 in DR,we get ‖e1(1− e1)f
′′ − 2e1f′‖r > 0.
Indeed, supposing the contrary it follows that z(1− z)f ′′(z)− 2zf ′(z) = 0 forall |z| ≤ r, that is (1− z)f ′′(z)− 2f ′(z) = 0 for all |z| ≤ r with z = 0. The lastequality is equivalent to [(1 − z)f ′(z)]′ − f ′(z) = 0, for all |z| ≤ r with z = 0.Therefore we get (1−z)f ′(z)−f(z) = C, with C a constant, that is f(z) = Cz
1−z ,
for all |z| ≤ r with z = 0.
But since f is analytic in Dr and r ≥ 1, we necessarily have C = 0 (contrari-wise, we would get that f(z) is not differentiable at z = 1, which is impossible),a contradiction with the hypothesis.
Now by Theorem 2, we have
n2
∥∥∥∥Mn(f, ·)− f − e1(1− e1)f′′ − 2e1f
′′
n
∥∥∥∥r
≤ Mr(f).
Therefore there exists an index n0 depending only on f and r, such that for alln ≥ n0, we have
‖e1(1− e1)f′′ − 2e1f
′‖r − 1
n
{n2
∥∥∥∥Mn(f, ·)− f − e1(1− e1)f′′ − 2e1f
′
n
∥∥∥∥r
}
≥ 1
2‖e1(1− e1)f
′′ − 2e1f′‖r,
which immediately implies
‖Mn(f, ·)− f‖r ≥ 1
2n‖e1(1− e1)f
′′ − 2e1f′‖r, for all n ≥ n0.
For n ∈ {1, 2, . . . , n0 − 1} we obviously have ‖Mn(f, ·) − f‖r ≥ Mr,n(f)n with
Mr,n(f) = n‖Mn(f, ·) − f‖r > 0. Indeed, since Mn(f, z) is a polynomial of
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APPROXIMATION BY COMPLEX SUMMATION-INTEGRAL TYPE OPERATOR
degree ≤ n, from the equality Mn(f, z) = f(z) for all |z| ≤ r, it necessarily
follows that f(z) is a polynomial of degree ≤ n. Let f(z) =n∑
k=0
akzk. We get
Mn(f, z) =
n∑k=0
akMn(ek, z) =
n∑k=0
akzk.
But by Lemma 1 (or by Lemma 2, (ii) ), it is clear that the coefficient of ekin Mn(ek, z) is equal to 1 only for k = 0. This necessarily implies that f is aconstant function, contradicting the hypothesis i.e. f is a polynomial of degree
> 0. Therefore finally we obtain ‖Mn(f, ·)− f‖r ≥ Cr(f)n for all n, where
Cr(f) = min{Mr,1(f),Mr,2(f), . . . ,Mr,n0−1(f),
1
2‖e1(1− e1)f
′′ − 2e1f′‖r
},
which completes the proof. �
As a consequence of Theorem 1 and Theorem 3, we have the following:
�������� 4� Let R > 1 and suppose that f : DR → C is analytic in DR. Iff is not a polynomial of degree zero, then for any r ∈ [1, R), we have
‖Mn(f, ·)− f‖r ∼ 1
n, n ∈ N,
where the constants in the equivalence depend only on f and r.
4. Applications
As a first application of the approximation properties of these Durrmeyer-typepolynomials, we can mention some shape preserving properties. Thus, reasoningexactly as in the case of complex Bernstein polynomials in [7], one can provethat beginning with an index, the Durrmeyer-type polynomials in the presentpaper approximate the analytic functions, preserving in addition,the classicalproperties of univalence, star likeness, convexity and spiral likeness in geometricfunction theory. Also, as a potential application, we can mention the possibil-ity to represent some C0-semigroups generated by a complex one-dimensionalsecond-order differential equation acting on the space of analytic functions inan open disk, as a limit of iterates of these complex polynomials, exactly as itwas done in the classical well-known case of positive linear operators acting onspaces of continuous functions of real variable.
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VIJAY GUPTA — RANI YADAV
Acknowledgement� The authors are thankful to Professor Sorin G. Gal forvaluable discussions and providing the necessary applications for overall im-provements of the manuscript. Thanks are also due to the reviewers for theirvaluable suggestion leading to overall improvements in the paper.
REFERENCES
[1] ABEL, U.—GUPTA, V.—MOHAPATRA, R. N.: Local approximation by a variant ofBernstein Durrmeter operators, Nonlinear Anal. 68 (2008), 3372–3381.
[2] ANASTASSIOU, G. A.—GAL, S. G.: Approximation by complex Bernstein-Durrmeyerpolynomials in compact disks, Mediterr. J. Math. 7(4) (2010), 471–482.
[3] GAL, S. G.: Shape Preserving Approximation by Real and Complex Polynomials,Birkhauser Publ., Boston, 2008.
[4] GAL, S. G.: Approximation by Complex Bernstein and Convolution-Type Operators,
World Scientific Publ. Co, Singapore-Hong Kong-London-New Jersey, 2009.[5] GAL, S. G.: Approximation by complex genuine Durrmeyer type polynomials in compact
disks, Appl. Math. Comput. 217 (2010), 1913–1920.[6] GAL, S. G.: Approximation by complex Bernstein-Durrmeyer polynomials with Jacobi
weights in compact disks, Math. Balkanica (N.S.) 24 (2010), 103–119.[7] GAL, S. G.: Voronovskajas theorem, shape preserving properties and iterations for com-
plex q-Bernstein polynomials, Studia Sci. Math. Hungar. 48 (2011), 23–43.[8] LORENTZ, G. G.: Bernstein Polynomials (2nd ed.), Chelsea Publ., New York, 1986.
Received 17. 4. 2011Accepted 20. 9. 2011
School of Applied SciencesNetaji Subhas Institute of TechnologySector 3 Dwarka New Delhi-110078
